Identification of the dynamic characteristics of nonlinear structures

CHAPTER 
IDENTIFICATION OF
NONLINEARITY USING 
ORDER FREQUENCY RESPONSE FUNCTIONS
2.1 GENERAL INTRODUCTION
As mentioned in Chapter 1, it is believed that all practical engineering structures are
nonlinear to some extent, due to nonlinearities in structural joints, boundary conditions
and material properties. Some structures may be only slightly nonlinear so that they can
be analysed based on a linear theory with satisfactory accuracy. For others whose
nonlinearities are such that their dynamic characteristics deviate considerably from linear
behaviour, nonlinear analysis techniques have to be employed.
As far as analyses of nonlinear systems are considered, there are two types of activity
which are loosely termed here as “theoretical analysis” and “experimental analysis”. The
theoretical analysis methods assume that a mathematical model of the nonlinear system to
be analysed is known (usually in the form of differential equations) and what is of interest
is the prediction of the response of the system due to a certain input. In contrast, the
experimental analysis methods seek ways of identifying mathematical models of nonlinear
systems based on measured input/output dynamic characteristics.
Since, in general, nonlinear problems do not possess closed form solutions, both
theoretical and experimental analyses are approximate and the accuracy of each analysis
depends on the mathematical nature of the problem and the specific methods employed.
What is of major interest in this thesis is the experimental identification of nonlinear

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Identification of Nonlinearity Using First-order 
12
structures based on structural modal testing, but since the theoretical analysis provides the
basis for the experimental identification of nonlinear structures, it is necessary to give a
brief introduction to the theoretical analysis of nonlinear systems.
Nonlinear systems with either inherent nonlinear characteristics or nonlinearities
deliberately introduced into the system to improve their dynamic characteristics have
found wide applications in most diverse fields of engineering. The principal task of
nonlinear system analysis is obtain a comprehensive picture, quantitative if possible, but
at least qualitative, of what happens to the system if it is driven into its nonlinear regime.
According to whether the system variables such as vibration displacement in the
mechanical structure are perturbed only slightly or largely from their operating points (for
most nonlinear mechanical structures, the nonlinear effect becomes more severe as the
vibration amplitude increases, but there are some exceptions such as friction
nonlinearity), the nonlinear characteristics can be divided into 
or 
global 
behaviour.
Local behaviour can be investigated by rather general and efficient linear methods that are
based on the powerful superposition principle as explained in Chapter 1 because, in this
case, the dynamic characteristics of the system are completely dominated by linear
behaviour. However, if these linear methods are extended to describe the global
behaviour of a nonlinear system, the results can be erroneous both quantitatively and
qualitatively since, in this case, the nonlinear characteristics may be essential but the linear
methods may fail to reveal it. Therefore, there is a strong emphasis on the development of
methods and techniques for the analysis and design of nonlinear systems.
However, it has to be mentioned that the development of nonlinear methods faces real
difficulties for a variety of reasons. There are no universal mathematical methods for the
solution of nonlinear differential equations which are the mathematical models of
nonlinear systems. The methods which exist deal with specific classes of nonlinear
equations and therefore have limited applicability to system analysis. The classification of
a given system and the choice of an appropriate method of analysis is not at all an easy
task. Furthermore, even in simple nonlinear problems, there are numerous new
phenomena qualitatively different from those expected in linear system behaviour, and it
is impossible to encompass all these phenomena in a single and unique method of
analysis.
Although there is no universal approach to the analysis of nonlinear systems, nonlinear
methods generally fall into one of the three following approaches: (i) the phase-space
topological method, (i) the stability analysis method, or (iii) the approximate method of
nonlinear analysis. These are summarised below.

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2 Identification of Nonlinearity Using First-order 
13
(i) The phase-space or, more specifically, the phase-plane approach has been used for
solving problems in mathematics and engineering at least since 
in 
The
approach gives both local and global behaviour of the nonlinear system and provides an
exact topological account of all possible system motions under various operating
conditions. It is a powerful concept underlying the entire theory of ordinary differential
equations (linear, nonlinear, time-varying and time-invariant). However, it is limited to
the case of second-order equations. For higher-order systems, this approach is very
cumbersome to use.
(ii) The stability analysis of nonlinear systems, which is heavily based on the work of
Lyapunov, is a powerful approach to the qualitative analysis of system global behaviour.
By this approach, the global behaviour of the system is investigated utilizing a given form
of nonlinear differential equations without explicit knowledge of their solution. Stability
is an inherent feature of a wide class of systems such as aerospace structures.
(iii) Approximate methods for solving problems in mathematical physics were first
developed at the beginning of this century. They have been received with much interest
by engineers and have promptly obtained wide application in diverse fields of
engineering. The basic merit of approximate methods lies in their being direct and
efficient and they permit a simple evaluation of the solution for a wide class of problems
arising in the analysis of nonlinear oscillations.
In the theoretical analysis of nonlinear systems whose equations of motion can be
formulated analytically, there are quite a number of approximate methods available to
examine their nonlinear vibration behaviour. According to different input signals,
methods in general can be 
into 
deterministic 
methods, in which the excitation
signals are deterministic such as sinusoids and 
statistical 
methods, in which the input
signal is of a random nature. Statistical analysis methods include the method(s) of random
linearisation 
and the amplitude domain analysis based on the FPK equations 
as discussed in Chapter 1. On the other hand, in deterministic analysis, the most
commonly used methods are the 
method 
the method of multiple
scale 
and the harmonic balance method 
What is of particular interest here is
the harmonic balance method (often called describing function method) because this
harmonic balance analysis provides the mathematical basis for a new nonlinear modal
analysis method developed in this chapter. The harmonic balance method is heavily based
on the Krylov-Bogoliubov approach 
and is applicable to nonlinear systems described
by higher-order differential equations. The mathematical basis of the method and the

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Identification of Nonlinearity Using First-order 
14
applicability conditions when it is used to identify the nonlinearities of mechanical
structures will be discussed in detail later in this chapter when the new nonlinear modal
analysis method is introduced. However, it will be appropriate here to introduce the
measurement techniques available in the dynamic 
of nonlinear structures
since what is of primary interest is the identification of nonlinear structures from the
actual test data.
2.2